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2020 Scaling limit of sub-ballistic 1D random walk among biased conductances: a story of wells and walls
Quentin Berger, Michele Salvi
Electron. J. Probab. 25: 1-43 (2020). DOI: 10.1214/20-EJP427


We consider a one-dimensional random walk among biased i.i.d. conductances, in the case where the random walk is transient but sub-ballistic: this occurs when the conductances have a heavy-tail at $+\infty $ or at $0$. We prove that the scaling limit of the process is the inverse of an $\alpha $-stable subordinator, which indicates an aging phenomenon, expressed in terms of the generalized arcsine law. In analogy with the case of an i.i.d. random environment studied in details in [ESZ09a, ESTZ13], some “traps” are responsible for the slowdown of the random walk. However, the phenomenology is somehow different (and richer) here. In particular, three types of traps may occur, depending on the fine properties of the tails of the conductances: (i) a very large conductance (a well in the potential); (ii) a very small conductance (a wall in the potential); (iii) the combination of a large conductance followed shortly after by a small conductance (a well-and-wall in the potential).


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Quentin Berger. Michele Salvi. "Scaling limit of sub-ballistic 1D random walk among biased conductances: a story of wells and walls." Electron. J. Probab. 25 1 - 43, 2020.


Received: 15 April 2019; Accepted: 31 January 2020; Published: 2020
First available in Project Euclid: 24 February 2020

zbMATH: 1441.60090
MathSciNet: MR4073691
Digital Object Identifier: 10.1214/20-EJP427

Primary: 60F17 , 60G52 , 60K37
Secondary: 82D30

Keywords: Aging , conductance model , Localization , random environment , Random walk , Scaling limit , sub-ballisticity , subordinator , trapping phenomenon


Vol.25 • 2020
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